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The debate over closing four Boston libraries will be reignited next week on Beacon Hill when lawmakers consider a proposal that makes an unmistakable threat to the city: Shutter a single branch and say goodbye to what is left of your state funding.

The threat, submitted as a cluster of budget amendments, would withhold the $2.4 million the Boston Public Library expects to receive from the state next year. If the amendments pass, the city, in order to receive the funding, would have to keep open all its 26 library branches in some capacity, regardless of staffing levels, layoffs, or hours of operation.

The idea, according to the 12 members of the Boston delegation spearheading the measures, is to keep each branch afloat until the economy rebounds and a proposed gambling bill perhaps refills state coffers with a gush of new tax revenue. Underscoring that point, the legislators are also seeking to block from Boston’s state aid what the city estimated as $500,000 in unrelated revenue from racetracks if any libraries close.

“The issue is the priorities of the city,’’ said Representative Martha M. Walz, a Democrat from the Back Bay, pointing to the city’s decision to give insurance company Liberty Mutual a $16 million tax break to expand its headquarters on Berkeley Street. “The mayor has chosen to prioritize other things over the libraries.’’

But the state has slashed its funding for the Boston Public Library by 73 percent over the last two years, from $8.9 million to a proposed $2.4 million next fiscal year. And the House budget proposal, unveiled last week, would not restore a cent.

Boston Public Library president Amy E. Ryan pushed back in a letter she sent last week in a point-by-point response to the lawmakers, writing that the current system is not sustainable.

“Propping up the status quo in this changing world is not working for today’s services or into the future,’’ Ryan wrote. “Instead, our budget decisions have positioned the Boston Public Library for financial stability and sustainability long into the future.’’

The proposal to withhold state funding is included in three of the 869 amendments tacked onto the budget that House members will begin debating next week.

Other amendments would restore some state funding to the Boston Public Library. (The state covers only a fraction of the library’s budget.) It is difficult to gauge how many lawmakers would support any of the individual measures.

“This amendment, like all others, will be subject to the collective will of the membership,’’ said Wayne Weikel, a spokesman for the House Committee on Ways and Means.

The library’s Board of Trustees voted April 9 to close four branches as part of a series of cuts that will lay off up to 77 people. Chairman Jeffrey B. Rudman chided the state delegation for not attending the meeting as the board discussed such an important decision.

At least one member made it to the library a few minutes that day, Representative Linda Dorcena Forry, a Democrat from Dorchester who has since signed on to the measures to block state funding.

“This is not about the money,’’ Forry said yesterday. “It’s about them wanting to closing libraries. I think this is four and then down the road who knows how many more.’’

Library officials have said that closing four branches will keep the remaining locations intact. The system was stretched too thin, Ryan said, and previous layoffs forced “brownouts’’ at certain branches, requiring closing so staff could eat lunch or dinner. And the cuts slashed the book budget and forced officials to scale back programming. But Ryan has also talked about transforming the library for the digital age.

“My gut is telling me that this has nothing to do with money,’’ said Representative Michael J. Moran, a Democrat from Brighton, who expressed anger that he did not get a personal phone call notifying him that one of the branches targeted for closing was in his district. “It has to do with the vision the president has for libraries. That’s really what this debate is about.’’